abstract

This study presents a direct methodology for the analysis of nonlinear dynamic systems with external periodic forcing via an application of the theory of normal forms. Rather than introducing a new state variable to reduce the problem to a homogenous one, we apply a set of time-dependant near-identity transformations to construct the normal forms. The proposed method can be applied to time-invariant as well as time varying systems. After discussing the time-invariant case, the methodology is extended to systems with time-periodic coefficients. The time periodic case is handled through an application of the Lyapunov-Floquet (L-F) transformation. It has been shown that all resonance conditions can be obtained in a closed form. Further, for time invariant case, if the superharmonic response is dominant, a simple modification can be made to yield accurate results. An example for each type of system, viz., constant coefficients and time-varying coefficients is included to demonstrate effectiveness of the method. It is observed that the linear parametric excitation term need not be small as generally assumed in perturbation and averaging techniques. The results obtained by proposed method are compared with numerical solutions.

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